Numerical calculations of annihilation radiation from point isotropic photon sources

Annals o/Nuclear Energy. Vol. 6. pp. 337 to 344

0306-4549'79 0601-0337502.00 0

Pergamon Press Ltd. 1979 Printed in Great Britain

NUMERICAL CALCULATIONS OF ANNIHILATION RADIATION FROM POINT ISOTROPIC PHOTON SOURCES G. GOUVRAS Nuclear Power Section. Department of Mechanical Engineering. Imperial College of Science and Technology. London SW7 2BX

(Received 23 November 1978; in revised.form 11 January 1979) Abstract--High energy photons give rise to annihilation radiation, especially in medium weight to heavy elements. A procedui'e is presented, based on the moments method, for calculating spectra of annihilation gamma rays and their contribution to exposure from point isotropic sources in three media: air, water and lead. It is shown that annihilation photons change the spectrum of scattered photons below 511 keV, but contribute little to exposure.

I. INTRODUCTION

2. THEORETICALCONSIDERATIONS

This study has arisen out of moments method calculations of the propagation of gamma rays in air from a point isotropic source, where a knowledge of the spectrum, particularly at low energies, was necessary for the calculation of human organ doses. In these calculations, pair production had been regarded as a purely absorbing process since in light media its probability of occurrence is very small compared to Compton scattering for initial photon energies up to 10MeV. The need to define the energy spectrum below 0.5 MeV, combined with the lack of comprehensive published information on annihilation contributions, led to the calculations reported here. Spectral distributions of annihilation photons from plane mono-directional sources in lead were given by Berger et al. 0959), while Dillman 0974) presented a method for computing fluxes of quanta from a uniform, infinitely extended radiation source in air. Goldstein (1954) has given estimates of the contribution to dose of annihilation radiation from plane mono-directional sources in various materials. This paper presents the results of a treatment of the point isotropic problem in three materials: air, water and lead. The program ENEMY was written to solve the photon transport equation using the moments method of Spencer arid Fano (1951). These moments can subsequently be used to calculate the additional annihilation fluxes because, due to the linearity of the transport process, the annihilation radiation is attenuated independently of the photons coming from the original source. Annihilation energy spectra, exposures and build-up factors for seven penetration distances and seven initial energies were computed using the code ANNIE developed using the method discussed below. p,.~.r 6 i 6 - (

337

Let I(L r') be the energy flux at r' from a point isotopic source located at the origin O (Fig. I), which is calculated ignoring annihilation radiation. Here, 2 is the Compton wavelength related to the photon energy by the equation 0.511 E(MeV) " An integration over tiae differential energy spectrum from 1.022 MeV to the initial energy, weighted by the probability of pair production per unit path length, yields the source function for pairs:

P(r', ~o) =

i

o.s d2'

T/~p(2')4nr"l(2', r').

(1)

The positron, generatedat any point on the spherical surface of radius r', will travel a distance r in the outward direction r'. This assumption of an outward directed positron may be justified because the photons which are producing pairs are confined mainly to a cone making an angle of less than 55 ° with r' and because the initial positron average obliquity is of the order of 00 = (mc2/E), where E is the

Fig. 1. Geometrical configuration.

338

G. GOUVRAS

energy of the p o s i t r o n ; E ' = mc 2 + ¢ if e is the initial p o s i t r o n kinetic energy. E x p e r i m e n t a l evidence s h o w s that, at low p h o t o n energies, the m e a n direction o f the p o s i t r o n lies within 15 ° o f the p h o t o n direction, whereas at high p h o t o n energies m o s t of the pair particles are e m i t t e d in the f o r w a r d direction (Segre, 1953). I n o r d e r to c o n f i r m the deflection o f initial p h o t o n s from their original line of travel extensive calculations of a n g u l a r distributions were m a d e using e x p a n s i o n s o f the energy flux into spherical harmonics. T h e b a c k g r o u n d for this p r o c e d u r e was provided by S p e n c e r (1952), b u t the p r e s e n t calculations involve, t h r o u g h the use o f a c o m p u t e r , a greater n u m b e r o f t e r m s in the L e g e n d r e e x p a n s i o n . S a m p l e results are p r e s e n t e d in Fig. 2. The average d i s p l a c e m e n t ~:(r', Eo) of a p o s i t r o n p r o d u c e d at r' is given by the e q u a t i o n

therefore, only a s e c o n d o r d e r influence o n the annihilation radiation correction. Accordingly, it was considered safe to a s s u m e that the p o s i t r o n s are annihilated at the p o i n t o f p r o d u c t i o n ; each then gives rise to t w o isotropically e m i t t e d p h o t o n s , each o f energy 511 keV, w h i c h travel in o p p o s i t e directions. W e disr e g a r d the possibility o f a n n i h i l a t i o n during the posit r o n flight b e c a u s e at energies b e l o w 10 M e V at least 80% o f all p o s i t r o n s in lead as well as air are annihilated w h e n at rest (Heitler, 1954). H u b b e l l (1968) estim a t e s the effect of a n n i h i l a t i o n in flight to a m o u n t to b e t w e e n 2 a n d 3% at 10 MeV. Therefore all annihilation p h o t o n s created at r' may be r e g a r d e d as forming an isotropic spherical surface source o f energy 0.511 MeV. If G(2, r; r') is the energy flux o f p h o t o n s at r due to a spherical source o f unit s t r e n g t h a n d o f radius r', emitting isotropically m o n o e n e r g e t i c p h o tons o f w a v e l e n g t h 2 = 1, t h e n the a n n i h i l a t i o n energy flux is given by

r(r',Eo)

H(;,,r;2o)= 2 fodr' f~i" d--~-,~,(2')4nr'Zlt,~',r')

f2E° d E _t, P~sin 0 dO I(E, cos 0, r')#p(E) cos 0 ( A ' ( E ) ) AV

l

eo

/-~ d E J, sinOdOl(E, cos0, r')gp(E)

2n~r

x G(2,r;r').

(3)

0

(2) J. METHOD OF SOLUTION

Values of (Ar(E))Av, the average d i s p l a c e m e n t o f a p o s i t r o n p r o d u c e d by a p h o t o n o f energy E, t o g e t h e r with the d e r i v a t i o n o f the e q u a t i o n , are given by Berger (1959). Results o f the full numerical treatm e n t o f the integrals are given in Table 1. It is seen that the average p o s i t r o n d i s p l a c e m e n t is quite small c o m p a r e d with a p h o t o n m e a n free p a t h a n d has,

C o n s i d e r the energy flux at r due to a plane isotropic source at the origin O (Fig. i), then iPt(~ r) = 2~

f;

p d p l ( ~ R),

(4)

from the figure, R 2 = p2 + r2; hence 2 p d p = 2 R d R ,

Table 1. Values of average displacement /~oz(/~r, Eo) of a positron produced at distance /~or by radiation originating from a point source with energy Eo Eo (MeV)

A/r 8 6 5 4 3

1

2

/~or 7

4

10

15

20

0.86 0.648 0.451 0.328 0.229

x x x x x

l0 -6 10 -6 10 -6 10 -6 10 -6

0.636 0.553 0.401 0.305 0.222

x x x x x

l0 -6 10 -6 10 -6 l0 -6 l0 -6

0.302 0.364 0.295 0.255 0.206

x x x x ×

10 -6 10 -6 10 -6 10 -6 10 -6

0.219 0.246 0.216 0.207 0.187

x x x x x

10 -6 10 -6 10 -6 10 -6 10 -6

0.211 0.225 0.199 0.192 0.177

x x x x x

10 -6 10 -6 10 -6 10 -6 10 -6

0.208 0.217 0.193 0.185 0.173

x x x x x

10 -6 10 .6 10 -6 10 -6 10 -6

0.208 0.215 0.191 0.183 0.172

x x x x x

10 -6 10 -6 10 -6 10 -6 10 -6

0.758 0.568 0.405 0.295 0.206

× x x x x

10 -3 10 -3 10 -3 10 -3 10 -3

0.552 0.480 0.359 0.274 0.200

x x x x x

10 -3 10 -3 10 -3 10 -3 10 -3

0.261 0.310 0.262 0.229 0.186

x x x x x

10 -3 10 -3 10 -3 10 -3 10 -3

0.193 0.213 0.192 0.186 0.168

x x x x x

10 -3 10 -3 10 -3 10 -3 10 -3

0.187 0.197 0.177 0.172 0.16

x x x x x

10 -3 10 -3 10 -3 10 -a 10 -3

0.185 0.191 0.172 0.167 0.156

x x x x x

10 -3 10 -3 10 -3 10 -3 10 -3

0.185 0.189 0.171 0.165 0.155

x x x x x

10 -3 10 -3 10 -3 10 -3 10 -3

Water 8 6 5 4 3

Lead 8 6 5 4 3

0.0185 0.0108 0.0066 0.0041 0.00246

0.0175 0.0103 0.0063 0.0039 0.00240

0.0155 0.0092 0.0056 0.0036 0.00226

0.0114 0.0072 0.0045 0.0030 0.00206

0.0077 0.0055 0.0036 0.0026 0.00194

0.0048 0.0040 0.0030 0.0024 0.00187

0.0042 0.0036 0.0028 0.0023 0.00185

Annihilation radiation from point photon sources ('0) I00

339

Air

'i $ 19 MeV

!

2.04 MeV

I 022 Me V

I0

I-~.J

I

\'\!

'\ '\!,

01

20

40

0.01

60

20

40

60

(b)

20

40

60

80

I00

Water

o

~oo~

,I

J ,!i 319 MeV

I0 I ~ '

2 0 4 MeV ,

1.022 MeV

/!

,

i

"\

x

ol x i

20

40

60

oo,

:

0

I

20

;

i

'~ ,

40

60

(C)

0

20

40

60

80

I00

Leod

IO0

to0

\:

319

MeV

'\\,, ~"" ~,

2 0 4 MeV

~,'~-"~" ~

/1

[.022 MeV

\

I

\ Ol

01

i

0

20

4C

60

o

i

I

2O Angle,

40

60

i

O0

C

20

40

60

80

degrees

Fig. 2. Angular distributions of photons in (a) air, (b) water and (c) lead from a point isotrop]c source emitting 1 photon of energy 8 MeV per second. Spectral components shown are for 3.19, 2.04 and 1.022 MeV. Solid curves represent fluxes at 1 mfp distance: broken curves are for a distance of 4 mfp, with the rest showing fluxes at 10 mfp. Ordinates have been multiplied by 4 ~ 2 e "°'

340

G. GOUVRAS

and therefore

revert to (9) and expand I(2' r') in Taylor series: ~x

lP~(2, r) = 27r J

R dR I(2, R).

l(2',r') = V (r' - r)" c7'I(2',r)

(5)

n=o

Consider now the energy flux at # due to the spherical surface array of point sources of radius r':

tl !

(~r n

(11) '

whereas I P' (Z r + r') is substituted by the expression 5

G(2, r;r') = 21tSa

(6)

r'2sinOdOl(2, R'),

IP'(2,r + r') ~ ½e -"'('÷"') S" Uk(2)Vk(/axr + plr'). k=0

(12) where So (the shell source strength per unit area) = 1/4nr '2. Since R a = r 2 + r '2 - 2rr' cos 0, it follows that 2R' dR' = 2rr' sin 0 dO, hence G(Zr;r') = ~ 1

2 rt r' r ~f,-v -,"r' R ' d R ' I O . , R ' ) .

(7)

The polynomials Uk are linear combinations of the plane isotropic moments and the Vk are used for the reconstruction of the flux from its moments (Fano et aL, 1959). Finally, the derivatives of the point isotropic flux are computed by an expression similar to (12). The result is

A comparison of (5) and (7) yields the relationship 1

2

~ I "°5d2'

&l(A',r)]

r'

G(2, r; r') = 4nr,---5 r [IP1(2' r - r') - lrt(2, r + r')]. (8) x

The flux H can now be expressed as H(2, r; 20) = 2

dr'

~

Ht - H:),

where r

Ht --- (n + 1)I~'+10.) + plrlP,~(2)

#p(Z)10.', r') -

o

r

+ (-- 1)"

x [Iel(2,r - r') - lel(Lr + r')].

H2 = ½e -~'' ~ bl(k,/~lr)

r'" g'le'(2, r) ( - 1)" ~. ~

IPl(2,r + r') =

n=O

(15)

r'" ~Ie'(2, r) n! Or" '

The coefficients bi in the expression of H2 in (15) are obtained from the polynomials Uk and Vk. Values of the linear coefficients for pair production. p~, were computed using cross section data from Knipe (1976). The integral

1 r°'5 d2' ~ 1 J ~ -;;-'/~v(Z) ~ 2n + 1 ~r o Z n~0 ×

( - 1)~(i + n - k)!

Or n

then (9) becomes H(2'r;2°) =

(14)

and

If we use the Taylor expansions

n=0

x"(x -- lqr)d.xlet(x) ir

(9)

IP'O.,r - r') =

(13)

(/,/1 ~ 2.+1 G2"I+ l/P'(~, r) -\~0/ 0(fllr)2n+ I 12n(~')'

(10)

where Po, #1 are the linear attenuation coefficients at Eo and 0.511 MeV, respectively, and uo2,+~ F~ 12"(2') = (2n)----~.Jo r""dr'4nr'2I(2 ',r')

ff

it

x"(x - i~lr)dxlV'(x)

(16)

is a small corrective term which becomes negligible for sufficiently large ~or. For a penetration of N mean free paths the lower limit of the integral is N

#lr = #1 #o

\/~o/

the point isotropic moments of the energy flux. EquaExamination of the ratio t~l/Po for the energy range tion (10) is an elegant formula for the annihilation and materials investigated in this work leads to the flux but an inadequate one because the series contains conclusion that (16) is only significant for fight materthe ratio #t/#o which is of the order of 2 to 3 for ials and lower energies. In lead, for example, the lower the materials considered in this report. Thus conver- limit is greater than 3 even for one mean free path gence of the series cannot be achieved. Instead we of penetration. A numerical approximation to the in-

Annihilation radiation from point photon sources tegral was included in the calculations; this could be done easily since the plane isotropic fluxes are computed using the same moments and polynomials utilized in the point isotropic problem. Finally, we note that in (14) the first term exists only for n odd, whereas the second only for n even, due to the symmetry of the plane isotropic source. In both cases the odd moments of the flux vanish.

341

4. RESULTS AND

DISCUSSION

Spectra of the annihilation energy flux in air, water and lead were computed for energies up to 8 MeV and for seven penetration distances. Results for 8 and 6 MeV and at 1, 4 and 10 mfp are shown in Figs 3 and 4. The ordinate scale is such that the fluxes are multiplied by a factor 4nr 2 e "°', so as to eliminate

(o)Air I0 i

i0 ~

I I0 0

-

lO-i

_

i0-- 3

~-~

I,','t

I0 -I

' ~

I-

~

, I0-'

I

.

r

10 0

I J,Y

~ O-- 3

I0 ~

, I0

?

F I,,._ "^\ ,o

"~J

I0 -z

U.

~or = I0

i0 c

,o. "

~

I'

\:,'~.

,e

I0 -2

4

I I0 -I

I0 0

I0 I

(b) Water ~

i0 r

I0 o

I0 c

,o-~- I,'~

I

,o-.L I'l' =

"~\'~"

I,'l'

,o-'

I

"~

10-2

II

'

r ' lO-r

I~;

lO I

~?- ¢

!

'- J

I

I) ~ I

I0-3 I0 0

I

I ~/

I

Eo" 8 MeV lO -~

I

i0-2

Eo= 6 MeV Irl'lH

) I0-I

i

111111 I00

q

~ i inlHJ lO I

Fig. 3. Energy spectra of photons in (a) air and (b) water. Broken curves represent annihilation fluxes which have been divided by ten for clarity of presentation. The total flux, i.e. the sum of annihilation and unperturbed flux, is also given for the energy region below 0.511 MeV. Ordinates have been multiplied by 4nr 2 e"°'.

342

G.

Gouve~ (a) E.. 8 MeV

i0 j

i01

i0 0

I°° I

j

i0-1

41

I ///i t /%r= I

_- 10-3rO-I

i00

IOJ

',

I

g 1

~O-1

iO O

tO ~

I0-'

I0'

~00

(b) Eo=6 MeV ~

,o~!

I0 q

j

~0o

IO0

I0-= /I

/ /

~0°I

/1111 I ~0-' -

/

]

/// i

lO-Z

IO'ZI

I

/~o r= 4

10-3 lO-i

I0 o

~0~

10-3 10 I

= i ,qiH

i

I0 o Energy, MeV

E

, , ,ill

i0 J

/~ ,"= lO

] 10-3 IO-I

~0o

I0'

Fig. 4. Energy spectra of photons in lead for (a) 8 MeV and (b) 6 MeV initial photon energy Eo. Broken curves represent annihilation fluxes. The total flux, i.e. the sum of annihilation and unperturbed flux is also shown, starting at 0.511 MeV. Ordinates have been multiplied by 4nr 2 e ~ ' . the steep inverse square and exponential dependence. Fluxes for the unperturbed problem, i.e. the point isotropic source without annihilation radiation are also given for comparison. For the light materials the annihilation component of the energy flux may be up to 50yo of the unperturbed component and has a similar variation with energy. In lead, however, the situation is completely different. The total flux has a very strong peak at 0.511 MeV due to the appearance of annihilation photons at that energy and below; this effect is more pronounced at smaller distances from the source and increases as the energy increases. This annihilation

peak is a well known feature of the photon spectra in heavy elements and comparisons with plane isotropic data (for example Bishop, 1978) could be made using this method. The presence of annihilation more than doubles the energy flux for short penetration depths in the energy region of 20)--511 keV, as predicted by Goldstein (1954). In Fig. 5 the total annihilation flux (including the unscattered component) is plotted against initial photon energy for various penetration depths. The curves for lead show a steady increase in the flux above 1.022 MeV, the energy threshold for pair production. In air and water the flux attains a maximum, the

Annihilation radiation from point photon sources

343 (b)watar

(O) Air iO-It

10"5 t

S

10-12

/~o r , l

S

10-6

E o IO-I~

Po r - I

10-7

>.

2

10-,4

tO-S

f

10-15

S 10-9 i

~J JO-IO

o I0-~6

10-=7

jO-m 0

,S I

2

t 4

3

5

6

7

I0

IO-Ul

8

jO-B2 o Energy,

if

i I

I0

2

J 3

pr.

I

I 4

£ 5

I 6

£ 7

P 8

MeV

(C) L e a d , 10-3

10-4

i0_5

~; 10-6 x 2 u.

lO_r

w

~ Io-8

10-9

io-tO O

! I

I 2

I t 3 4 Energy,

,, .L 5 MeV

I 6

I 7

t 8

9

Fig. 5. Total annihilation energy flux as a function of the initial photon energy for #or = 1, 4, 10.

location of which depends on the material. In air this maximum occurs around 6 MeV, whereas in water the m a x i m u m is shifted towards 5 MeV. This behaviour is explained from the pair production pattern in the three materials, represented by (1). An evaluation of the integral yields the pair production rate per unit path length, which reveals the source maxima in the case of water and air; or the steady increase

with energy as in lead. Clearly the rise in the value of pair production coefficient above the 1.022 MeV threshold is offset by a simultaneous decrease in the value of attenuation coefficient in the case of air or water; in lead, however, the attenuation coefficient increases with energy above the broad m i n i m u m at about 3.4 MeV, so that a steady growth in pair production is observed.

344

G. GouvaAs

Table 2. Percentage increase in exposure due to the presence of annihilation radiation Material

Energy (MeV)

l

2

4

#o r 7

10

15

20

Air

8 6 4

1.07 1.19 1.39 1.61 1.73 1.87 1.94 1.02 1.12 1.28 1.43 1.52 1.62 1.67 0.'85 0.88 0.94 0.98 1.01 1.04 1.06

Water

8 6 4

1.03 1.14 1.28 1.50 1.63 1.76 1.83 0.96 1.04 1.16 1.30 1.39 1.48 1.53 0.80 0.82 0.85 0.88 0.91 0.94 0.96

Lead

8 6 4

1.5l 1.75 2.18 2.64 3.03 3.21 3.13 1.40 1.64 2.06 2.56 2.85 3.00 2.96 1.23 1.41 1.71 2.10 2.31 2.44 2.46

Contributions of the annihilation radiation to exposure are shown in Table 2. U p o n integrating the annihilation p h o t o n spectra, weighted by the mass absorption coefficients of air, the total exposure rate is obtained; this is subsequently divided by the total exposure rate of the u n p e r t u r b e d spectra and the results are given in Table 2 as percentage increases of the exposure. For all three materials the contribution is greater in higher energies, with values in lead being significantly higher at respective initial energies than those i n air and water. Examination of the dependence on distance shows an initial increase in the contribution, followed by a levelling off at larger distances. In air and water this process starts a r o u n d 20 mfp or even earlier in lower energies. In lead the m a x i m u m is reached a r o u n d 15 mfp at 8 and 6 MeV, but in lower energies it is shifted towards greater distances. The percentage increase at large distances in water and lead seems to confirm Goldstein's upper limits for the point isotropic problem (Goldstein. Table 3. Annihilation radiation exposure build-up factors Energy (MeV)

1

2

4

/aor 7

10

15

20

Air

8 6 5 4

3.36 3.48 3.58 3.72

3.33 3.44 3.54 3.68

3.28 3.38 3.47 3.59

3.21 3.30 3.38 3.50

3.17 3.25 3.31 3.42

3.14 3.21 3.26 3.36

3.13 3.19 3.24 3.33

Water

8 6 5 4

3.60 3.78 3.91 4.08

3.50 3.66 3.80 3.95

3.36 3.48 3.60 3.75

3.23 3.33 3.4l 3.55

3.19 3.27 3.34 3.46

3.17 3.23 3.30 3.39

3.15 3.22 3.28 3.37

Lead

8 6 5 4

1.27 1.27 1.26 1.26

1.25 1.25 1.25 1.25

1.23 1.23 1.23 1.23

1.22 1.22 1.22 1.22

1.22 1.22 1.22 1.22

1.22 1.22 1.22 1.22

1.22 1.22 1.22 1.22

Material

1954). Lower values at shorter distances are expected when c o m p a r i n g these results with plane perpendicular source geometry (Berger, 1959), because the ratio of the point flux to plane perpendicular flux is less than 1 for small penetration depths. An interesting feature of the calculations are the annihilation radiation build-up factors, given in Table 3. They demonstrate a lack of sensitivity to distance and, in the case of lead, to energy. This, if the a t t e n u a t i o n coefficient at the initial energy /~o is at all less than the a t t e n u a t i o n coefficient at 0.511 MeV (/al), is intrinsically expected, because, as shown by Goldstein (1954), the exposure from an exponentially distributed source is very similar to that from a uniformly distributed source, in which case the ratio of the total to the unscattered flux is constant t h r o u g h o u t the medium. In conclusion, the presence of annihilation radiation was shown to have little effect on the energy spectra a n d resulting exposure in light media. In air the effects on the energy flux can be safely assumed negligible, with the contribution to exposure being less t h a n 2?/o. In heavy elements, by contrast, the annihilation contributes considerably to the p h o t o n spect r u m below 0.511 MeV. However, its c o n t r i b u t i o n to exposure becomes somehow significant only in energies above 6 MeV. Acknowledgement--The author wishes to thank Dr A~ J. H. Goddard, his research supervisor, whose critical comments constitute the conditio sine qua non of this paper. REFERENCES

Berger M. J., Hubbell J. H. and Reingold 1. H. (1959) Phys. Rev. 113, 857. Bishop G. B. and Marafie A. M. (1978) Ann. nucl. Enerqy 5, 35. DiUman L. T. (1974) Health Phys. 2"1, 571. Fano U., Spencer L. V. and Berger M. J. (1959) Handh. Phys. 38, 744, Springer. Berlin. Goldstein H. (1954) Nuclear Development Associates Report NDA-15C-31. Heitler W. 11954) The Quantum Theory of Radiation, p. 384, Oxford University Press, London. Hubbell J. H. and Berger M. J. (1968) Engineerin 9 Compendium on Radiation ShieldinO Vol. 1 p. 180. Springer, New York. Knipe A. D. (1975) A Revision of Photon Interaction Data in the UKAEA Nuclear Data Library, AEEW-MI368, Winfrith, Dorset. Segr/: E. (ed.) (1953) Experimental Nuclear Physics Vol 1, p. 335, Wiley, New York. Spencer L. V. and Fano U. (t951) Bur. Stand. J. Res. 46, 441. Spencer L. V. and Stinson F. (1952) Phys. Rev. 85, 662.